The aggregation of cell surface receptors triggers transmembrane signals in a variety of biological systems. In the immune system the aggregation of cell surface immunoglobulin triggers both "on" and "off" signals on B lymphocytes, basophils, and mast cells. Our long range goal is to quantitatively understand such signals, and in so doing understand how antigens in allergic reactions activate and desensitize basophils and mast cells. Our approach is to study the interaction of simple multivalent DNP-haptens, first with monoclonal anti-DNP immunoglobulin E (IgE) in solution, then on vesicles, and finally on rat basophilic leukemia (RBL) cells at 4 C and 37 C. Using mathematical models, we will analyze binding studies carried out by our collaborators at Cornell. For a series of bivalent DNP-haptens of different length and flexibility we will determine their hapten-IgE binding, cross-linking, and ring closure constants. Knowing the values of these constants will allow us to calculate the entire hapten-IgE aggregate size distribution. We will then investigate how these binding constants and the predicted aggregate distribution correlate with the haptens' ability to activate and desensitize basophils. We hope to learn, for example, why some bivalent haptens that appear to bind well to sensitized basophils are very poor at causing basophilic degranulation. We also will use DNP-haptens of valence 3 and 4 to further study the role of receptor aggregation in basophil signaling. Finally, once we have fully characterized the binding of our haptens to cell surface IgE, we will study the interaction of these haptens with sensitized RBL cells at 37 C where aggregate IgE appears to be internalized. Again, through interaction of experiment and mathematical modeling, we hope to gain insight into the cell surface events that occur prior to and during internalization.